Rings and Integral Domains

• September 16th 2008, 10:13 AM
Aryth
Rings and Integral Domains
Let $\mathcal{Z}[x]$ be the set of all polynomials in the variable x, with coefficients from $\mathcal{Z}$ and the usual operations of addition and multiplication.

Let $S$ be the set of polynomials in $\mathcal{Z}[x]$ with roots at $x=0$ and $x=1$. Prove that $S$ is a subring of $\mathcal{Z}[x]$.
• September 16th 2008, 10:37 AM
ThePerfectHacker
Quote:

Originally Posted by Aryth
Let $\mathcal{Z}[x]$ be the set of all polynomials in the variable x, with coefficients from $\mathcal{Z}$ and the usual operations of addition and multiplication.

Let $S$ be the set of polynomials in $\mathcal{Z}[x]$ with roots at $x=0$ and $x=1$. Prove that $S$ is a subring of $\mathcal{Z}[x]$.

Do it by definition.
For example, for additive closure say $f(x),g(x) \in S$.
Then $f(x)+g(x) \in S$ since $f(0)+g(0) = 0 + 0 = 0$ and $f(1)+g(1) = 0 + 0 = 0$.